A turbulent-entropic instability and the fragmentation of star-forming clouds

The kinetic energy of supersonic turbulence within interstellar clouds is subject to cooling by dissipation in shocks and subsequent line radiation. The clouds are therefore susceptible to a condensation process controlled by the specific entropy. In a form analogous to the thermodynamic entropy, the entropy for supersonic turbulence is proportional to the log of the product of the mean turbulent velocity and the size scale. We derive a dispersion relation for the growth of entropic instabilities in a spherical self-gravitating cloud and find that there is a critical maximum dissipation time-scale, about equal to the crossing time, that allows for fragmentation and subsequent star formation. However, the time-scale for the loss of turbulent energy may be shorter or longer, for example, with rapid thermal cooling or the injection of mechanical energy. Differences in the time-scale for energy loss in different star-forming regions may result in differences in the outcome, for example, in the initial mass function.